Study information

# Number Theory - 2023 entry

MODULE TITLE CREDIT VALUE Number Theory 15 MTH3004 Dr Christopher Lazda (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 weeks 0 0
 Number of Students Taking Module (anticipated) 100
DESCRIPTION - summary of the module content

Number theory is a vast and fascinating field of mathematics, consisting of the study of the properties of whole numbers. From this module, you will acquire a working knowledge of the main concepts of classical elementary number theory. This will be developed as a rigorous proof-based theory along with some appreciation of the theory behind modern computational techniques. Topics studied include divisibility properties of natural numbers, congruences, prime numbers, primality and factorisation, quadratic reciprocity, sums of squares and Fermat’s last theorem for the special case of sums of fourth powers.

Prerequisite module: MTH2002 or MTH2010, or equivalent

AIMS - intentions of the module

This course covers one of the oldest and most popular areas of mathematics, building on basic ideas and including modern applications. The dual objectives are to provide a solid foundation for further work in number theory, but also at the same time to give a self-contained interesting course suitable as an end in itself, with modern answers to ancient problems and modern applications of classical ideas. You will acquire a sound foundation in number theory from a modern perspective.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)
On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:
1 demonstrate a working knowledge of the main concepts of classical elementary number theory, together with some appreciation of modern computational techniques.

Discipline Specific Skills and Knowledge:
2 understand the role of Number Theory as a central topic in mathematics, and demonstrate an awareness of some of its modern applications;

Personal and Key Transferable/ Employment Skills and Knowledge:
3 show enhanced problem-solving skills and ability to formulate your solutions as mathematical proofs;
4 reveal a fundamental knowledge of Number Theory from a modern perspective.

SYLLABUS PLAN - summary of the structure and academic content of the module
• divisibility, greatest common divisor;
• extended Euclidean algorithm, prime numbers and unique factorisation;
• modular arithmetic, Euler's and Wilson's theorems, Chinese Remainder Theorem;
• polynomial congruences, Hensel lifting;
• primitive roots;
• sums of two and four squares;
• Pythagorean triples;
• Fermat's Last Theorem for exponent four.
LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled Learning and Teaching Activities 33 Lectures/example classes Guided Independent Study 117 Guided independent study

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework – example sheets Variable All Written and verbal

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 20 80 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework 1 – based on questions submitted for assessment 10 15 hours All Annotated script and written/verbal feedback
Coursework 2 - based on questions submitted for assessment  10 15 hours All Annotated script and written/verbal feedback
Written Exam - closed book 80 2 hours (Summer) All Written/verbal on request, SRS

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
Written Exam* Written Exam (2 hours) All August Ref/Def Period
Coursework 1* Coursework 1 All August Ref/Def Period
Coursework 2* Coursework 2 All August Ref/Def Period

*Please refer to reassessment notes for details on deferral vs. Referral reassessment

RE-ASSESSMENT NOTES

Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only.  For deferred candidates, the module mark will be uncapped.

Referrals: Reassessment will be by a single written exam worth 100% of the module only.  As it is a referral, the mark will be capped at 40%.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk